This invention relates to the field of frequency response function calculation, and more particularly to the field of stimulus signal generation for frequency response function calculations with maximum dynamic range.
A frequency response function describes the relationship between the input and output of a physical system. The system involved can be either mechanical, electrical, or some other type of linear physical system. To characterize such systems, it desirable to be able to accurately calculate their frequency response function, which in most cases is a very close approximation of the actual transfer function of the system.
The frequency response function is complex, since for each frequency component of the input to the system, there is both a magnitude value and phase value to the system's response.
A function, x(t), describes the magnitude of the stimulus signal to the system as a function of time. The Fourier transform function, X(f), of the stimulus signal, x(t), describes the magnitude and phase content of the stimulus signal as a function of frequency. Similarly, another function, y(t), describes the magnitude of the output from the system as a function of time. And, there is a related Fourier transform function, Y(f), that describes the magnitude and phase of the output signal, y(t), as a function of frequency. EQU X(f)=Fourier Transform of x(t) (1) EQU Y(f)=Fourier Transform of y(t) (2)
If the stimulus and output functions of a system, x(t) and y(t), are measured, the corresponding Fourier Transforms, X(f) and Y(f), can be calculated using a Fast Fourier Transform (FFT) algorithm or other similar method. Then, the frequency response function, H(f), can be calculated according to the following relationship: ##EQU1## where, Cspec(X,Y) is the cross power spectrum between X(f) and Y(f), Aspec(X) is the auto power spectrum of Y(f), and H(f) is the frequency response function of the system, which in most cases is a good estimation of the actual transfer function of the system, H(f).
Modern instruments, such as the 2642 Personal Fourier Analyzer made by Tektronix, Inc., Beaverton, Oreg., provide a means for supplying stimulus to a system and measuring its response. The computational facilities and other capabilities of this instrument are described in the 2641/2642 Fourier Analyzer User's Guide and the 2641/2642 Fourier Analyzer TurboPac Application Library, both of which are hereby incorporated by reference.
Among the relevant capabilities and computational facilities available in the 2642 Personal Fourier Analyzer are the ability to perform the Fast Fourier Transform (FFT), the Inverse Fourier Transform (IFT), and the ability to calculate the frequency response function, H(f), between two signals on its inputs. The latter computation necessarily entails performing the cross power spectrum (Cspec) between the stimulus and output channels and the auto power spectrum (Aspec) on the stimulus channel.
Despite the automation of all of these computational facilities in this instrument, there are limitations to the calculations that can be made that arise from its inherent limitations. Specifically, there is a limit to the dynamic range of the frequency characteristics that can be accurately measured by each input channel. The dynamic range of a channel refers to the overall ability of the instrument to distinguish between real signals and those artifacts and distortions that inevitably occur during measurement and computation. These include such factors as the non-linearity of analog components and A/D converters, signal leakage through power supplies and logic signals, jitter in the sampling clock, aliasing products, and truncation and other errors in the arithmetic associated with digital filtering, FFT computations, and averaging operations.
The dynamic range specification for the 2642 Personal Fourier Analyzer is 75 dB. This means that if a known-to-be-pure full-scale sine wave is applied to one of the inputs of the Analyzer, all of the associated spectral artifacts will be below the full scale by at least 75 dB. But, no matter how good the dynamic range of the input channels may be, there is still always some such limit.
One of the original methods of characterizing the frequency response function of a system under analysis, one that does not require powerful FFT analyzers, is to provide a pure sinusoidal signal at one frequency at the input, and then to measure the output of the system at that frequency at the output. By slowly varying the frequency, and measuring the system output at each frequency, the system can eventually be characterized for all frequencies of interest. This approach requires a lot of time, but does not require a lot of dynamic range on the input and output channels, since the settings of the input and output channels can be varied from frequency to frequency as the sweeping of the bandwidth of interest occurs.
Another approach, one that is preferred by many because it characterizes the whole bandwidth of interest very rapidly, is to apply broadband noise with a flat power spectrum, i.e., white noise, such that the Aspec of X(f) approaches a constant, to the system under analysis over the bandwidth of interest. The spectrum measured at the output of the system is then a close approximation of the system's estimated frequency response function, H(f), directly.
A variation on the foregoing method is too anticipate, by estimation or rough prior calculation, the system response and provide a signal source that is the inverse of the estimated frequency response function, 1/ H(f). The system output function, Y(f), will then be approximately a constant. But, to the extent that it is not a constant, the standard calculation of the transfer function can proceed, per equation (3) above, and the results can be used to refine the input function estimate and the process repeated.
From the point of view of the dynamic range limitations of the Fourier analyzer being used to make these measurements, both of these approaches suffer from the same major limitation. In each case, one channel, either the input channel or the output channel, has to supply most of the dynamic range required to make the measurement. If the input power spectrum is flat, the full variation of the frequency response function appears in the output spectrum, while if the output power spectrum is made flat the full range of the inverted frequency response function appears in the input spectrum.
What is desired is a method for generating a stimulus signal for conducting frequency response function calculations that maximizes the overall dynamic range of the calculations possible by balancing the requirements for dynamic range between the two analyzer channels being used to make the calculation.